The investigation of subterranean formations is a common occurrence in oil and gas exploration and production operations. Methods and tools for investigating subsurface formations have advanced considerably over the years. There are many commercially available acoustic, nuclear, electromagnetic, and resistivity tools that provide a variety of information about formations adjacent to a borehole.
Early electrical methods of exploration in the 1920s tested electrical resistivity and electrostatic potential, but proved to be more successful at locating metallic ores than oil and gas. Oil and gas have resistivity or conductivity properties that differ from water, which conducts electricity more readily. Occurrences of oil and gas can sometimes be located by this difference in conductivity. Conductivity measurements also indicate porosity and hydrocarbon saturation, which is a very important component of petrophysical resource assessment.
The usefulness of the conductivity measurements is dependent on the accuracy of the true formation conductivity (Ct). Apparent formation conductivity (Ca), as measured by a standard logging tool, however, is not equal to true formation conductivity (Ct) in most logging environments because of the limitations of tool physics and non-ideal borehole conditions. Deep-reading conductivity/resistivity tools cannot resolve formations less than a few feet thick, and cannot make accurate true-conductivity (Ct) measurements when the borehole diameter is variable or when drilling mud or other fluids with a different resistivity than the formation fluids has seeped into the formation (invasion), thereby altering the conductivity of the invaded zone (Cxo).
The traditional method of correcting the environmental effects on the accuracy of conductivity measurements has been the use of chartbooks provided by logging service companies. However, chartbooks only contain a limited number of charts with strict assumptions (e.g., borehole diameter, mud resistivity, and Ct/Cxo ratio) that rarely match real world examples. Therefore, chartbook corrections may only serve to make a qualitative estimation. Furthermore, the nonlinear conductivity tool response (due to borehole diameter, mud conductivity, invasion, and bed thickness or shoulder bed effects all together) can not be corrected from the chartbooks' corrections without assumptions of linear superposition.
Computer inverse modeling of resistivity tool response can be conducted to convert apparent resistivity from logs into a response profile that may closely approximate reality. In fact, modern environmental correction charts provided by service companies are the result of computer forward modeling. In general, the inverse modeling involves replicating the observed field log by numerically solving the mathematical boundary value problems of the electrical or electromagnetic fields generated by a specific resistivity tool under a predefined layered-earth model. To the degree that the field log and the computed tool response are in acceptable agreement through iterative forward modeling, the underlying earth model may be considered as one possible representation of the formation's true resistivity or conductivity profile. Mathematically, such an inversion process attempts to fit the computed tool response under a set of earth parameters (e.g., bed thickness, Ct, Cm, Cxo, borehole diameter and invasion depth) to an actual field conductivity log, or a set of actual field logs. The parameters in the earth model can be refined by solving least-squares problems through the iterative process to minimize the sum of the squares of the errors between the computed tool response and the measured field log. The iteration may continue until the fit between the computed and field logs reaches predetermined criteria.
With the advent of modeling codes and the significant increase in computing power, resistivity tool response modeling has become a feasible option for formation evaluation. 2D inversion of resistivity logging tool measurements based on iterative 2D forward modeling with finite element or hybrid methods are described in Gianzero, S., Lin, Y., and Su, S., 1985, “A new high speed hybrid technique for simulation and inversion of resistivity logs”, SPE 14189; Liu, Q., H., 1994, “Nonlinear inversion of electrode-type resistivity measurements”, IEEE on Geoscience and Remote Sensing, Vol. 32, No. 3, pp 499-507; and Mezzatesta, A. G., Eckard, M. H., Strack, K. M., 1995, “Integrated 2D Interpretation of Resistivity Logging Measurements by Inversion Methods”, Paper-E, SPWLA 36th Annual Logging Symposium Transactions.
However, the tool response or sensitivity calculation used to update an iterative inversion model takes time to compute. Traditional analytic computation of sensitivity is linear with respect to the number of parameters. The number of parameters used for an accurate resistivity log often reaches the hundreds, significantly slowing the computation time.
The present invention is directed to overcoming, or at least reducing the effects of, one or more of the problems outlined above.